Abstract: In a jonit work with Weisheng Wu, we propose a new method for finding solutions of cohomological equations via periodic cycle functionals. Instead of considering all cycles, we focus on a much smaller class of homotopically trivial cycles. It is shown that, vanishing of periodic cycle functionals arising from homotopically trivial cycles is a complete obstruction to obtaining a $C^\infty$ solution for a cohomological equation, for each one of the following systems,
(1) ergodic affine endomorphisms on torus which has a non-trivial contracting bundle,
(2) partially hyperbolic automorphisms on Heisenberg nilmanifolds,
(3) partially hyperbolic flows on certain homogeneous spaces (under an additional condition).
We also obtain results concerning solutions with intermediate regularities.